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In a recent work of Duke, Imamo\={g}lu, and T\'{o}th, the linking number ofcertain links on the space$\text{SL}(2,\mathbb{Z})\backslash\text{SL}(2,\mathbb{R})$ is investigated.This linking number has an alternative interpretation as the intersectionnumber of closed geodesics on the modular curve, which is the focus of thispaper. By relating the intersection number to a combinatorial computationinvolving rivers of Conway topographs, an efficient algorithm for computingintersection numbers is produced. A formula for the total intersection of apair of positive coprime fundamental discriminants is also derived, which canbe thought of as a real quadratic analogue of a classical result of Gross andZagier. The paper ends with numerical computations and distribution questionsrelating to intersection numbers.

Computing intersections of closed geodesics on the modular curve